Thermal Uhlmann Chern number from the Uhlmann connection for extracting topological properties of mixed states

TL;DR

This paper introduces the thermal Uhlmann Chern number, a topological invariant for mixed quantum states that extends the concept of topological characterization to finite-temperature systems, using the Uhlmann connection.

Contribution

It proposes a modified Chern character to define a thermal Uhlmann Chern number that captures topological properties of mixed states, overcoming the triviality of the original Uhlmann connection.

Findings

The thermal Uhlmann Chern number reflects the topology of the Hamiltonian.

It distinguishes topological phases at finite temperature.

Application to two-band and four-band models demonstrates its effectiveness.

Abstract

The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well defined global section. This property implies that the Uhlmann connection is topologically trivial and as a consequence, the corresponding Chern character vanishes. We propose modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the non-topological thermal Uhlmann Chern number which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.